Firstly, we use the techniques of Ref. [1]} to
derive bounds between low-energy couplings of the same dimensionality
in gravitational scattering.It would be
very interesting to generalize our analysis to include bounds that
relate couplings of different dimensionality along the lines of
Refs. [2]}, [3]}, [4]}.
We focus on the first few corrections to Einstein gravity. We then ask
where do the Wilson coefficients obtained from string theory and from
the low-energy limit of the one-loop minimally-coupled amplitudes land
in the space allowed by the general bounds. Remarkably, in all cases
studied here we find that both the string and field-theory
coefficients land on a small theory island, which to a good
approximation is a thin line segment in the space of EFT
coefficients. (See, for example, Fig. REF  and
Fig. REF  in
Sect. ).In
Ref. [5]} the string-theory island was interpreted
in terms of unitarity constraints coupled with world-sheet monodromy
constraints. The location of this island can be found by assuming
that lowest-spin partial waves dominate the dispersive representation
of the low-energy couplings, which is the LSD principle. See Sect. REF  for the
precise mathematical formulation. More generally, we show how one can
combine an assumed hierarchy among the spectral densities of various
spins with crossing symmetry to systematically derive stronger bounds
on the Wilson coefficients. We impose crossing symmetry via the use
of null constraints [2]}, [3]}.
